Scilab Function
Last update : 12/1/2004

binomial - binomial distribution probabilities

### Calling Sequence

  `pr=binomial(p,n)`

### Parameters

- `pr`: row vector with n+1 components
- `p`: real number in [0,1]
- `n`: an integer >= 1

### Description

`pr=binomial(p,n)` returns the binomial probability vector, i.e. `pr(k+1)` is the probability of `k` success in `n` independent Bernouilli trials with probability of success `p`. In other words : `pr(k+1) = probability(X=k)` , with X a random variable following the B(n,p) distribution, and numerically :

                / n \  k      n-k       / n \       n!
     pr(k+1) =  |   | p  (1-p)    with  |   |  = ---------
                \ k /                   \ k /    k! (n-k)!

### Examples

    // first example
    n=10;p=0.3; xbasc(); plot2d3(0:n,binomial(p,n));

    // second example
    n=50;p=0.4;
    mea=n*p; sigma=sqrt(n*p*(1-p));
    x=( (0:n)-mea )/sigma;
    xbasc()
    plot2d(x, sigma*binomial(p,n));
    deff('y=Gauss(x)','y=1/sqrt(2*%pi)*exp(-(x.^2)/2)')
    plot2d(x, Gauss(x), style=2);

    // by binomial formula (Caution if big n)
    function pr=binomial2(p,n)
    x=poly(0,'x');pr=coeff((1-p+x)^n).*horner(x^(0:n),p);
    endfunction
    p=1/3;n=5;
    binomial(p,n)-binomial2(p,n)

    // by Gamma function: gamma(n+1)=n! (Caution if big n)
    p=1/3;n=5;
    Cnks=gamma(n+1)./(gamma(1:n+1).*gamma(n+1:-1:1));
    x=poly(0,'x');
    pr=Cnks.*horner(x.^(0:n).*(1-x)^(n:-1:0),p);
    pr-binomial(p,n)

### See Also

`cdfbin`, `grand`,